Generic SSD drive keeps disconnecting during Catalina install or back up

It is hard to say as the issue could be any of the items you referenced or it could be a cable issue or a port issue with your laptop (hardware or software).

Is the drive connected directly to the laptop? Many times hubs & docks are poorly made. If the SSD has been connected directly to the laptop have you tried connecting the drive to a good brand of powered USB3 hub?

What is the make & model of the SSD?

How are you connecting the drive to your Mac? Have you tried a different cable? A different port on the laptop? Have you tried the drive with another computer?

Did you erase the whole physical drive instead of just erasing the volume/partition?

To check the health of the SSD you can try running DriveDx, but you will need to install a special USB driver so that DriveDx can access the external drive's health information. Even with the special USB driver DriveDx may not be able to access the SSD's health information because some external drives' USB chipsets don't allow the necessary communication. If you can access the SSD's health report, then post it here using the "Additional Text" icon which looks like a piece of paper.

You may want to return the SSD if the eBay seller has a return policy.

We've seen a number of posts from people who bought computers or disk drives on eBay and ended up with equipment that did not work properly. It's something of a gamble. The problem is, for something like a disk drive, the reason a person is selling it on eBay could well be that it was intermittently not working ... Unless you can find a simple solution like a bad cable or a bad enclosure/connector, I don't think I would ever trust that drive for anything crucial.

We use MANY external drives here and when they start disconnecting like that unexpectedly, we erase/format them and discard them (both mechanical and SSD). A small percentage of them do have these issues. Troubleshooting them is often a long process without satisfactory resolution. They are mass produced and the manufacturers accept a certain percentage of failure rate within their business model. A lower failure rate would either raise the price of the drive or result in lower profits. (You can almost tell what that failure rate is by looking on Amazon and seeing the number of "one star" reviews, it's often on the order of 1% or so.) But a used device from eBay in a generic enclosure will probably have a higher chance of failure.

As HWTech indicated, running DriveDX will give you specific info about how much of the drive's life has been used up, based on measured SMART parameters and some other diagnostics. But the device does not seem usable in its present form. (Hoping you did not pay a lot ...)

Eek, eBay seller in China selling you a new drive but it came in a "generic" or questionable enclosure? I feel like someone has robbed you here; maybe you can have some recourse with eBay as someone sold you something that is supposed to be new but flat out does not work. Do the terms of the sale give you any right to file a grievance?

Going that route might be a hassle, especially if the price was low. If you have a different, trusted enclosure, you could try that, you sound pretty handy at moving drive hardware around. Or consider ordering a high quality enclosure from OWC, they have a good reputation for enclosures (for mechanical drives or SSDs). Good luck with this!